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The extrinsic Schwarzschild solution is a vacuum solution, meaning that it is valid for regions of spacetime where there is no matter or energy. This seems to imply that the Schwarzschild solution is only valid outside the spherical body. For inside the spherical object, the internal Schwarzschild solution is used, where the stress energy tensor is no longer 0. However, when we use kruskal coordinates, the coordinate system covers the inside of the spherical object (usually taken to be a black hole). How can the kruskal coordinates accurately describe interior region of black hole when it is only derived for when there’s a vacuum?

The question linked in the comments does not have an answer to this question, because the answers say that there can be spacetime curvature in a vacuum solution. My question is NOT “how can there be spacetime curvature in a vacuum?”. The question is why do we talk about the spacetime INSIDE a black hole (which supposedly has a mass) as Schwarzschild?

Qmechanic
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2 Answers2

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However, when we use kruskal coordinates, the coordinate system covers the inside of the spherical object (usually taken to be a black hole).

The interior (by which I mean, $r<r_S$) of a Schwarzschild black hole is also a vacuum, insofar as the stress-energy tensor vanishes there.


But it’s a black hole..doesn’t it have mass/energy?

The Schwarzschild metric is a vacuum solution to the Einstein equations. It describes the metric in a vacuum ($T_{\mu\nu}=0$) under the assumptions that the metric is spherically symmetric and stationary. If you model a star as a spherical fluid with a sharp boundary with pure vacuum outside it, then the exterior region would be described by the Schwarzschild metric while the interior of the star (where $T_{\mu\nu}\neq 0$) would be described by the interior Schwarzschild metric. In this case, the parameter $M$ which appears in the exterior Schwarzschild solution is the mass of the star.

One could ask, however, what happens when you assume that $T_{\mu\nu}=0$ everywhere. The most general spherically symmetric, stationary vacuum solution to Einstein's equations is

$$\mathrm ds^2 = -\left(1-\frac{\alpha}{r}\right)c^2\mathrm dt^2 + \left(1-\frac{\alpha}{r}\right)^{-1} \mathrm dr^2 + r^2 \mathrm d\Omega^2$$

where $\alpha$ is an unknown constant. If $\alpha = 0$, then this is just the standard Minkowski spacetime from special relativity. However, it's entirely possible that $\alpha \neq 0$, provided that we remove the set of points with $r=0$ from the spacetime manifold (otherwise, the metric would be undefined there).

One might object immediately by noting that for $r=\alpha$, the metric component $g_{rr}$ is undefined. However, this is what is called a coordinate singularity; it's not actually a problem with the spacetime, just with this choice of coordinates. One can switch to the Kruskal coordinate system to demonstrate this fact. In contrast, the singularity at $r=0$ is a genuine one, and so that set of points must be removed.

So if $\alpha \neq 0$, then what is it? Comparison with the exterior Schwarzschild solution outside of a star provides a physical interpretation - this is what you'd expect to see if a total mass $M\equiv \alpha c^2/2G$ was crushed down to a single point in space (which, to avoid infinities, must be excised from the spacetime manifold). However, it's important to note that at all points (other than the singularity, which again has been removed), $T_{\mu\nu}=0$. This is true both for $r>\alpha$ and $0<r<\alpha$.

[...] from observations black holes are not a single point. Supermassive black holes have a radius of ~15 million miles.

The 15 million miles you refer to is the event horizon of the black hole - the $\alpha$ which appears in my explanation above. Again, for a Schwarzschild black hole, $T_{\mu\nu}=0$ both outside the event horizon and inside the event horizon.

[...] For these black holes there is surely mass and energy, so it seems like we cannot speak of geodesics “inside” the black hole if we are still sticking with schwartzchild space time

One of the reasons I keep using the qualifier Schwarzschild to refer to a black hole is because this metric is the simplest possible model for understanding the phenomenon. It is not necessarily a realistic model for an actual black hole, which would generically be e.g. rotating and surrounded by an accretion disk which is slowly falling below the event horizon. Rather, you should think of it as a toy model for understanding the nuances of the event horizon and other highly non-intuitive phenomena. Adding a steady influx of mass or some other more complex features would only obscure these issues.

J. Murray
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  • But it’s a black hole..doesn’t it have mass/energy? –  Jan 21 '23 at 23:13
  • @Obama2020 Nope. – TimRias Jan 22 '23 at 00:09
  • @TimRias Then what does the “m” in the schwartzchild metric stand for? Mass of what? Why do we say that black hole mass decreases due to Hawking radiation if it has no mass? –  Jan 22 '23 at 01:21
  • @PeterShor from observations black holes are not a single point. Supermassive black holes have a radius of ~15 million miles. For these black holes there is surely mass and energy, so it seems like we cannot speak of geodesics “inside” the black hole if we are still sticking with schwartzchild space time –  Jan 22 '23 at 01:39
  • @Obama2020 I have added a substantial addendum to my answer to address your questions. – J. Murray Jan 22 '23 at 02:34
  • @J.Murray I have a problem with $T_{\mu \nu}=0$ everywhere. Einstein should have said: “It was formerly believed that if all material things disappeared out of the universe, time and space would be left. According to relativity theory, however, time and space disappear together with the things.” https://rloldershaw.medium.com/einstein-without-matter-there-is-no-space-or-time-c2357c75286b. What would be your answer? – JanG Jan 22 '23 at 15:31
  • @JanGogolin My answer to what? There's nothing wrong with setting the right-hand side of the Einstein equations to zero. The existence of vacuum solutions to the Einstein equations is fairly obvious, many of which are non-trivial. That quote was delivered to an audience of reporters as a media sound bite, not a technical explanation of relativity. It's also worth noting that Einstein himself did not believe that black holes could form, so his work is not a particularly good place to look to understand them. – J. Murray Jan 22 '23 at 15:58
  • @J.Murray. Spacetime without matter is maybe mathematically correct but physically an oxymoron. Who would ask and answer? We are matter! – JanG Jan 22 '23 at 17:13
  • @JanGogolin Do you approach electromagnetism with the same spirit? Do you have a problem with setting $(\rho,\mathbf J )= 0$ in Maxwell's equations to study the propagation of light in the absence of sources? When studying projectile motion in physics 101, do you demand to include air resistance, the curvature of the earth, and the gravitational attraction of the projectile to the observer in your calculations? – J. Murray Jan 23 '23 at 00:39
  • @JanGogolin It's true that there is no such thing as a perfect vacuum, so setting $(\rho,\mathbf J)=0$ in Maxwell's equations is never physically correct. However, it simplifies the mathematics enormously while also capturing many of the salient features of electromagnetic waves. The model can be made more realistic from there by considering waves in linear dielectrics, and more realistic still by modeling a frequency dependent permittivity/permeability, but all of those extensions are based on a toy model which is gradually expanded. The GR case is, in this instance, precisely analogous. – J. Murray Jan 23 '23 at 00:43
  • @J.Murray I am afraid I cannot agree with you. I can imagine spacetime with dark matter only without electromagnetic fields and charges. The other way round, spacetime without gravitation but with electromagnetic fields and charges, in my understanding, does not exist. Minkowski flat spacetime is for me just a spacetime with numerically negligible curvature. – JanG Jan 23 '23 at 09:08
  • @JanGogolin You are of course welcome to your opinion. My point is simply that setting $T_{\mu\nu}=0$ is precisely analogous to setting $(\rho,\mathbf J)=0$ in that it is (obviously) not a particularly realistic model, but it is the basis of an important class of interesting solutions. – J. Murray Jan 23 '23 at 14:16
  • @J.Murray I appreciate your comments, thanks! – JanG Jan 23 '23 at 15:37
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Schwarzschild solution is valid only outside of matter's sphere and it is static - there is no time dependence. In case of black hole it ends on the even horizon. Behind it, time and space change their place and the resulting metric ("extended" Schwarzschild) is no more static - it depends explicitly on time. Hence, the black hole spacetime have two regions, separated by the event horizon, that are described by two different metrics. The use of the name Schwarzschild for both of them is somewhat misleading but due to history. Schwarzschild entitled his paper "On the gravitational field of a mass point according to Einstein's theory". He was not aware of the immense meaning of his result and the fact that in general relativity there is no something like a "mass point".

JanG
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  • Your answer is correct +1, but you statement on Schwarzschild being "unaware" of the meaning of his solution is not. In his paper, he states that the radius of the horizon defined as $r=\sqrt{x^2+y^2+z^2}$ equals zero, which is completely true, and therefore a black hole is indeed a mass point of a zero spatial radius. In the modern definition, the radius of the horizon is not zero, but its value is timelike, not spacelike. This means that the "distance" from the horizon to the origin is a not a distance in space, but a period of time. Thus spatially a black hole still is a mass point. – safesphere Jan 23 '23 at 06:44
  • @safesphere I have to admit your last sentence I find difficult to accept. Please look into https://physics.stackexchange.com/a/682496/281096 where similar question and explanations have been given. Is there something there you would agree? – JanG Jan 23 '23 at 08:58
  • The spatial Schwarzschild radial distance from the horizon to the origin is zero. As we observe from outside, the volume inside the horizon is zero as well: https://arxiv.org/abs/0801.1734 - It may seem hard to accept, because the popular propaganda has been painting a wrong picture for decades, but mathematics proves Schwarzschild right. There is nothing in the quote from t’Hooft that contradicts Schwarzschild. To get from the horizon to the singularity, you don’t need to move, just wait. You don’t need to change your location - the spatial location of the horizon and singularity is the same. – safesphere Jan 23 '23 at 15:43
  • @safesphere I am with you. The spatial location of the event horizon and the "singularity", where pressure diverges by constant energy density, is the same. I believe to have proved it for the case of perfect fluid spheres. – JanG Jan 23 '23 at 15:54